Theorem nosgnn0 33165

Description: ∅ is not a surreal sign. (Contributed by Scott Fenton, 16-Jun-2011.)

Assertion

Ref	Expression
nosgnn0	⊢ ¬ ∅ ∈ {1o, 2o}

Proof of Theorem nosgnn0

Step	Hyp	Ref	Expression
1		1n0 8119	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3073	. . 3 ⊢ ¬ ∅ = 1o
3		nsuceq0 6271	. . . . 5 ⊢ suc 1o ≠ ∅
4		necom 3069	. . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5		df-2o 8103	. . . . . . 7 ⊢ 2o = suc 1o
6	5	neeq2i 3081	. . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
7	4, 6	bitr4i 280	. . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
8	3, 7	mpbi 232	. . . 4 ⊢ ∅ ≠ 2o
9	8	neii 3018	. . 3 ⊢ ¬ ∅ = 2o
10	2, 9	pm3.2ni 877	. 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o)
11		0ex 5211	. . 3 ⊢ ∅ ∈ V
12	11	elpr 4590	. 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
13	10, 12	mtbir 325	1 ⊢ ¬ ∅ ∈ {1o, 2o}

Syntax hints:  ¬ wn 3   ∨ wo 843   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∅c0 4291  {cpr 4569  suc csuc 6193  1_oc1o 8095  2_oc2o 8096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-nul 5210

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-v 3496  df-dif 3939  df-un 3941  df-nul 4292  df-sn 4568  df-pr 4570  df-suc 6197  df-1o 8102  df-2o 8103

This theorem is referenced by:  nosgnn0i  33166  sltres  33169  noseponlem  33171  sltso  33181  nosepssdm  33190  nodenselem8  33195  nolt02olem  33198